# Question

Use the transformation-of-variable technique to prove Theorem 6.7 on page 188.

Theorem 6.7

If X has a normal distribution with the mean µ and the standard deviation s, then

Z = X – µ / σ

Has the standard normal distribution.

Theorem 6.7

If X has a normal distribution with the mean µ and the standard deviation s, then

Z = X – µ / σ

Has the standard normal distribution.

## Answer to relevant Questions

Use the transformation technique to rework Exercise 7.2. In exercise If the probability density of X is given by And Y = X2 If the probability density of X is given by And Y = X2, find (a) The distribution function of Y; (b) The probability density of Y. With reference to Example 3.12 on page 82, find (a) The probability distribution of U = X + Y; (b) The probability distribution of V = XY; (c) The probability distribution of W = X – Y. Rework Exercise 7.34 by using Theorem 7.2 to determine the joint probability density of U = Y – X and V = X and then finding the marginal density of U. If n independent random variables have the same gamma distribution with the parameters α and β, find the moment-generating function of their sum and, if possible, identify its distribution.Post your question

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